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## G = (C2×C8)⋊6F5order 320 = 26·5

### 4th semidirect product of C2×C8 and F5 acting via F5/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — (C2×C8)⋊6F5
 Chief series C1 — C5 — C10 — D10 — C4×D5 — C4⋊F5 — D10.C23 — (C2×C8)⋊6F5
 Lower central C5 — C10 — C20 — (C2×C8)⋊6F5
 Upper central C1 — C4 — C2×C4 — C2×C8

Generators and relations for (C2×C8)⋊6F5
G = < a,b,c,d | a2=b8=c5=d4=1, ab=ba, ac=ca, dad-1=ab4, bc=cb, dbd-1=b3, dcd-1=c3 >

Subgroups: 442 in 114 conjugacy classes, 50 normal (38 characteristic)
C1, C2, C2 [×4], C4 [×2], C4 [×6], C22, C22 [×4], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×9], C23, D5 [×2], D5, C10, C10, C42 [×2], C22⋊C4 [×2], C4⋊C4 [×4], C2×C8, C2×C8 [×5], C22×C4, Dic5 [×2], C20 [×2], F5 [×4], D10 [×2], D10 [×2], C2×C10, C4.Q8 [×2], C2.D8 [×2], C42⋊C2 [×2], C22×C8, C52C8 [×2], C40 [×2], C4×D5 [×4], C2×Dic5, C2×C20, C2×F5 [×4], C22×D5, C23.25D4, C8×D5 [×4], C2×C52C8, C2×C40, C4×F5 [×2], C4⋊F5 [×4], C22⋊F5 [×2], C2×C4×D5, C40⋊C4 [×2], D5.D8 [×2], D5×C2×C8, D10.C23 [×2], (C2×C8)⋊6F5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, F5, C2×C4⋊C4, C4○D8 [×2], C2×F5 [×3], C23.25D4, C4⋊F5 [×2], C22×F5, C2×C4⋊F5, (C2×C8)⋊6F5

Smallest permutation representation of (C2×C8)⋊6F5
On 80 points
Generators in S80
(1 26)(2 27)(3 28)(4 29)(5 30)(6 31)(7 32)(8 25)(9 69)(10 70)(11 71)(12 72)(13 65)(14 66)(15 67)(16 68)(17 47)(18 48)(19 41)(20 42)(21 43)(22 44)(23 45)(24 46)(33 63)(34 64)(35 57)(36 58)(37 59)(38 60)(39 61)(40 62)(49 78)(50 79)(51 80)(52 73)(53 74)(54 75)(55 76)(56 77)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80)
(1 68 49 46 59)(2 69 50 47 60)(3 70 51 48 61)(4 71 52 41 62)(5 72 53 42 63)(6 65 54 43 64)(7 66 55 44 57)(8 67 56 45 58)(9 79 17 38 27)(10 80 18 39 28)(11 73 19 40 29)(12 74 20 33 30)(13 75 21 34 31)(14 76 22 35 32)(15 77 23 36 25)(16 78 24 37 26)
(1 32 5 28)(2 27 6 31)(3 30 7 26)(4 25 8 29)(9 54 34 47)(10 49 35 42)(11 52 36 45)(12 55 37 48)(13 50 38 43)(14 53 39 46)(15 56 40 41)(16 51 33 44)(17 65 75 60)(18 68 76 63)(19 71 77 58)(20 66 78 61)(21 69 79 64)(22 72 80 59)(23 67 73 62)(24 70 74 57)

G:=sub<Sym(80)| (1,26)(2,27)(3,28)(4,29)(5,30)(6,31)(7,32)(8,25)(9,69)(10,70)(11,71)(12,72)(13,65)(14,66)(15,67)(16,68)(17,47)(18,48)(19,41)(20,42)(21,43)(22,44)(23,45)(24,46)(33,63)(34,64)(35,57)(36,58)(37,59)(38,60)(39,61)(40,62)(49,78)(50,79)(51,80)(52,73)(53,74)(54,75)(55,76)(56,77), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80), (1,68,49,46,59)(2,69,50,47,60)(3,70,51,48,61)(4,71,52,41,62)(5,72,53,42,63)(6,65,54,43,64)(7,66,55,44,57)(8,67,56,45,58)(9,79,17,38,27)(10,80,18,39,28)(11,73,19,40,29)(12,74,20,33,30)(13,75,21,34,31)(14,76,22,35,32)(15,77,23,36,25)(16,78,24,37,26), (1,32,5,28)(2,27,6,31)(3,30,7,26)(4,25,8,29)(9,54,34,47)(10,49,35,42)(11,52,36,45)(12,55,37,48)(13,50,38,43)(14,53,39,46)(15,56,40,41)(16,51,33,44)(17,65,75,60)(18,68,76,63)(19,71,77,58)(20,66,78,61)(21,69,79,64)(22,72,80,59)(23,67,73,62)(24,70,74,57)>;

G:=Group( (1,26)(2,27)(3,28)(4,29)(5,30)(6,31)(7,32)(8,25)(9,69)(10,70)(11,71)(12,72)(13,65)(14,66)(15,67)(16,68)(17,47)(18,48)(19,41)(20,42)(21,43)(22,44)(23,45)(24,46)(33,63)(34,64)(35,57)(36,58)(37,59)(38,60)(39,61)(40,62)(49,78)(50,79)(51,80)(52,73)(53,74)(54,75)(55,76)(56,77), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80), (1,68,49,46,59)(2,69,50,47,60)(3,70,51,48,61)(4,71,52,41,62)(5,72,53,42,63)(6,65,54,43,64)(7,66,55,44,57)(8,67,56,45,58)(9,79,17,38,27)(10,80,18,39,28)(11,73,19,40,29)(12,74,20,33,30)(13,75,21,34,31)(14,76,22,35,32)(15,77,23,36,25)(16,78,24,37,26), (1,32,5,28)(2,27,6,31)(3,30,7,26)(4,25,8,29)(9,54,34,47)(10,49,35,42)(11,52,36,45)(12,55,37,48)(13,50,38,43)(14,53,39,46)(15,56,40,41)(16,51,33,44)(17,65,75,60)(18,68,76,63)(19,71,77,58)(20,66,78,61)(21,69,79,64)(22,72,80,59)(23,67,73,62)(24,70,74,57) );

G=PermutationGroup([(1,26),(2,27),(3,28),(4,29),(5,30),(6,31),(7,32),(8,25),(9,69),(10,70),(11,71),(12,72),(13,65),(14,66),(15,67),(16,68),(17,47),(18,48),(19,41),(20,42),(21,43),(22,44),(23,45),(24,46),(33,63),(34,64),(35,57),(36,58),(37,59),(38,60),(39,61),(40,62),(49,78),(50,79),(51,80),(52,73),(53,74),(54,75),(55,76),(56,77)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80)], [(1,68,49,46,59),(2,69,50,47,60),(3,70,51,48,61),(4,71,52,41,62),(5,72,53,42,63),(6,65,54,43,64),(7,66,55,44,57),(8,67,56,45,58),(9,79,17,38,27),(10,80,18,39,28),(11,73,19,40,29),(12,74,20,33,30),(13,75,21,34,31),(14,76,22,35,32),(15,77,23,36,25),(16,78,24,37,26)], [(1,32,5,28),(2,27,6,31),(3,30,7,26),(4,25,8,29),(9,54,34,47),(10,49,35,42),(11,52,36,45),(12,55,37,48),(13,50,38,43),(14,53,39,46),(15,56,40,41),(16,51,33,44),(17,65,75,60),(18,68,76,63),(19,71,77,58),(20,66,78,61),(21,69,79,64),(22,72,80,59),(23,67,73,62),(24,70,74,57)])

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G ··· 4N 5 8A 8B 8C 8D 8E 8F 8G 8H 10A 10B 10C 20A 20B 20C 20D 40A ··· 40H order 1 2 2 2 2 2 4 4 4 4 4 4 4 ··· 4 5 8 8 8 8 8 8 8 8 10 10 10 20 20 20 20 40 ··· 40 size 1 1 2 5 5 10 1 1 2 5 5 10 20 ··· 20 4 2 2 2 2 10 10 10 10 4 4 4 4 4 4 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + - - + + + + image C1 C2 C2 C2 C2 C4 C4 C4 D4 Q8 Q8 D4 C4○D8 F5 C2×F5 C2×F5 C4⋊F5 C4⋊F5 (C2×C8)⋊6F5 kernel (C2×C8)⋊6F5 C40⋊C4 D5.D8 D5×C2×C8 D10.C23 C8×D5 C2×C5⋊2C8 C2×C40 C4×D5 C4×D5 C2×Dic5 C22×D5 D5 C2×C8 C8 C2×C4 C4 C22 C1 # reps 1 2 2 1 2 4 2 2 1 1 1 1 8 1 2 1 2 2 8

Matrix representation of (C2×C8)⋊6F5 in GL6(𝔽41)

 32 23 0 0 0 0 9 9 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 11 11 0 0 0 0 15 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 40 40 40 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 40 0 0 0 0 0 1 1 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 40 0 0 0 0 1 1 1 1

G:=sub<GL(6,GF(41))| [32,9,0,0,0,0,23,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[11,15,0,0,0,0,11,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,1,0,0,0,0,40,0,1,0,0,0,40,0,0,1,0,0,40,0,0,0],[40,1,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,1,0,0,0,0,40,1,0,0,0,0,0,1,0,0,0,40,0,1] >;

(C2×C8)⋊6F5 in GAP, Magma, Sage, TeX

(C_2\times C_8)\rtimes_6F_5
% in TeX

G:=Group("(C2xC8):6F5");
// GroupNames label

G:=SmallGroup(320,1059);
// by ID

G=gap.SmallGroup(320,1059);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,422,100,1684,102,6278,1595]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=c^5=d^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^4,b*c=c*b,d*b*d^-1=b^3,d*c*d^-1=c^3>;
// generators/relations

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